The effects of rotation on the rate of mass loss for O and B stars has been reviewed, and the causes for conflicting results discussed

Vardya, M. S.

English

Publications of the IAS Fellows

Bull Astr Soc lnc11.i (1990) 18, 309-313 
Rotation and mass loss in early type stars 
M .  S.  Vardya 
Tora J t ~ s r i r ~ r f e  oj F~ultkrtltrnral Re~rort l l .  f for lr i  Bhuhhu Koutl, 6011lhu1 400 005 
Abstract. The effect of rotation on the rate of mas5 loss for 0 and B stars 
has been rev~eu'ed. and the causes of confl~ct~ng results discussed. 
Ke) .  11~or~1.s . rotation--mass loss--early type stars 
Rotation decreases the effective surface gravity of a star, thereby decreasing the escape 
velocity. Thus one expects that rotation should enhance the rate of stellar mass loss df. 
Theoretically, de Greve, de Loore & de Jager ( 1  972) showed that the rate, of mass 
loss increases by 26 to 40:; for llnear rotation veloclty I- u p  to 200 km s-' in a F2 V type 
star. Marlborough & Zamir (1984) modified the CAK theory (Castor, Abbott & Klein 
1075) and showed that rotation increases the rate of mass loss over no-rotation value, 
)ugh they did not give any numerical value. 
What about the observational evidence? Furenlid & Young (1980) cons~dering 60 
rmal main sequence BO-B3 stars (excluding Be and peculiar stars) found that Ha l ~ n e  
rmmetry, which gives a measure of mass loss, is always large when projected h e a r  
ational velocity v sin i 2 200 km s-', however, they did not consider & itself and no 
-inite trend between asymmetry and v sin I is visible. Snow (1981) analysed 22 B stars 
om B0.5 to  B6) including 19 Be-like stars, Doazan er al. (1982) 21 Be, B shell and 
rmal stars, and Slettebak & Carpenter (1983) 12 Be and standard stars but found no 
nclusive evidence for rctation enhancing rate of mass loss. Gathier, Lamers & Snow 
181) did note a qualitative dependence of A? on v sin i in 25 high luminosity 0 and B 
rs, but found null result for early B superglants. 
The questlon naturally arises: Why this lack of definitiveness? Does the answer lie in 
: fact that observations provide projected rotational velocity v sin i when the theory 
mands v itself, without the aspect angle effect? And if lt is so, how we can circumvent 
A large mixed sample of stars should randomize the effect of sin i ,  making it possible 
discern the effect of rotation in spite of large individual deviations from the mean due 
the geometrical effect. All the above mentioned studies had considered not only a very 
- 
lited number of stars, but in a rather small range of spectral class. Therefore I (Vardya 
85) decided to consider a large sample of stars covering a large range in projected 
tational velocity, luminosity, temperature and rate of mass loss with the hope that sin i 
11 be randomized as well as possible. A total of 81 stars-49 from Lamers (1981), 14 
)m Garmany er 01. (1981) and 18 from Snow (3982)-were considered covering a range 
temperature spectral class from 0 3  to  B9, luminosity class la' to V, v sin i from, 15 to 
505 km s" and M from 3 X lo-'' to 2 X lo-' hft9 ,I-'. T h ~ s  ample has 21 0 stars w ~ t h  f, (f) 
and ((0) spectral characteristics, 10 Be stars or 18 Be llke stars, and s ~ x  pecul~ar stars, 
with a range of log L /  L+, from 2.53 to 6.38, Af/A.I,., from 7 to 136, and N /  Ha from 5 to 
86. This is a fairly rn~xed sample, though not a completely unb~;lsed onc. Note that all the 
18 stars from Snow (1982) appear to have Be-like prope~tles Irom the polnt of view of 
rate of mass loss In fact, the mean value of k for these He-l~kc star!, IS about tour orders 
of magnitude less than the rest of the 63 0 and B stars, wh~ch I~cicalttr we w~ll call 
normal OB stars 
No relatlon was found between the rate 01 mass loss and \, sln r but the rntc of mass 
loss per unlt area, I . ( .  , mass flux was found to correlate w~ th  sln r 1 hrrc were two 
separate relations-. one for the 63 normal OH stars and another one lor the 18 Hc-l~ke 
stars. Note that the mean value of mass flux for OB stars d~ffer from that ol I3c-l1ke stars 
by three orden of rnagn~tude. 
Though thls was encouraging, i t  was not sat~sfactory. Kotatlon, In a way, 1s an 
extrinsic property rherefore, gettlng two separate relat~ons rather than one was 
somewhat puzzling. Furthermore, rate of mass loss is four orders of magnitude less or 
mass flux three orders of magnitude less tor Be-ltke stars relat~vt: to normal OH stars, 
when the mean v sln i is three tlmes larger for Be-l~ke stars relative to normal O H  stars; 
this implies that other causes of mass loss dom~nate over rotation. W ~ t h  a hope of 
achiemng one slngle relation for both groups of stars, it was decided to subtract the 
dominant cause of mass loss I P radiation pressure. Th~s was done by using a relation 
that I (Vardya 1984) had found earlier for 0 and B stars, uslng dimens~onal and physlcal 
arguments. 
where L, R, and M are stellar luminos~ty, radlus, and mass, and A a scallng constant. 
Therefore, we considered a relat~on between A or rather A /  R' and 11 sin r .  Note that A 
may contaln the dependerce of not only rotation but of other parameters as  well, not 
considered so far, like magnetlc field. Thls resulted in a s~ngle relation for all the 8 1 stars, 
with a high correlation coefficient The correlat~on improved markedly when the 
projected linear rotational veloc~ty v sin i was replaced by the projected angular 
rotational velocity, fl sin i. Thus, 1 showed for the first t~me,  from observed data, that 
rotation definitely enhances the rate of mass loss, or more accurately, mass flux, 
confirming the theoretical expectations. 
Now, the question is, 1s the amount of enhancement commensurate with theoretical 
predictions? I had found from observations that A /  R increases by about 2.5 dex for an  
~ncrease of 1.5 dex in v sin i or 2.75 dex in ll sin i. Theoretically, an increase by - 26% in 
dl has been found as v goes from 0 to 350 krn s-' in a 05V star by Pauldrach, Puls & 
Kudritzki (1986); Poe & Friend have found for a 06ef star an enhancement of 62% in A? 
as v varies from 125 to 400 km s-I (with a magnetic field of 200 G), and an increase of 
370% for a B1.5Ve star as v goes from 125 to 540 km s-I (with a magnetic field of 50 G). 
Friend & Abbott (1986) have found that &I (rotation)/ fi (no-rotation) increases from 1 
to 2 as v (rotation)/ v (break-up) goes from 0 to  0.8; however, their final conclusion, using 
observationai data for 0 and B stars but excluding Be stars, is that there is currently no 
evidence for a dependence of the mass loss rates on rotational velocity, and the scatter in 
the observations is so large that it m a y  not be possible to find such a correlation even if it 
exists. And not to have any conflict with their own conclusions, Friend & Abbot (1986) 
Rotation and mass loss in early type stars 31 1 
have made this cryptic statement: "A correlation between mass-loss rate and rotational 
velocity has been sought by Vardya (1985), but the evidence is very weak at  best", 
without advancing any reason. 
Nieuwenhuijzen & de Jager (1988) have discirssed t h ~ s  difference between the 
theoretical conclusions and my (Vardya 1985) results, by considering 142 non-emission 
early type stars, but excluding Be and shell stars. According to them, the strong 
dependence that 1 found is a manifestat~on that both the rate of mass loss as well as 
v sin 1 1s correlated to the luminosity of the star, thus giving an artificial correlat~on 
between M and v sin i. Then they have fitted the d a b  of dl In terms of three variables, 
Tk,  L and v sin i by Chebychev polynomials of 39 coefficients, 20 independent of v sin I 
and 19 dependent on it, and have concluded that fi depends only slightly on v sin i. 
Critically examining their conclusions, we find that 
(a) Nieuwenhuijzen & de Jager (1988) have excluded Be and shell stars, thus 
preventing proper randomization of sin i. 
(b) In the sample that I (Vardya 1985) had used, the luminosity L is not correlated 
with I.? sin I ,  except in a limited region. In fact luminosity increases gradually wlth v sin i, 
reaches a peak around 150 km s-' and then decreases rather steeply..As a further check, a 
plot of log &I vs log v sin i at a given luminosity log L = 5.0 t 0.2, containing 23 stars, 
covering a range .of log M from -7.70 to -5.36 and of v sin i from 15 to 385 km s-' 
shows no correlation. Furthermole, 1 have differenced out the effect of luminosity by 
considering A rather than A?. 
(c) Chebychev polynomial fit of 39 coefficients-22 positive and 17 negative-with 
two-third coefficients of the same order of magnitude, may be a good numerical fit over a 
limited domain, but its utility ends there. Using it for physical interpretation is 
dangerous, to say the least. Besides, we are interested in v and not in v sin i dependence. 
And by such an accurate fit not only sin i has been incorporated but dependence of 
magnetic field as well. 
(d) One should note that if the sample is restricted, i.e. limited to a small range in 
s,pectral class for example, the scaling or constant factor will absorb similar dependence, 
thus preventing an explicit manifestation of real dependence. In  a similar way, when a 
large varied sample is fitted with an expression with a large number of coefficients, the 
real dependence will get absorbed in these coefficients and one will see only a kind of 
residual dependence. 
(e) When Nieuwenhuijzen & de Jager (1988) considered Be and shell stars, which I 
have included in my analysis, they found that mass loss rate ii larger by two order of 
ni;~pn;luds from the equatorial areas relative to high latitude parts, which was similar to 
other stars. 
(f) Theoretical results of Poe & Friend (1986) clearly show that the gradient of the 
increase of dl as v sin i increases, increases sharply as the critical velocity is approached. 
Recently Howarth & Prinja (1989) have considered 163 galactic 0 stars with v sin i 
between 5 to 435 km s-', log j$f between -4.6 to  -7.8, log L/Lo between 4.5 to 6.4, 
M /  Me between 18 to 150, and R/ Re between 5 to 36. They have found a change A log 
M 1 0 . 4  when the velocity goes from slowest to the fastest rotation; however their 
expression is not valid for v sin i G 153 km s-l. They have also like us (Vardya (1985) 
considered a quantity similar to our A, in which the effect of luminosity and luminosity 
class is eliminated. Note that though they have taken a large number of stars, it is 
restricted in spectral class, v sin i, log i@ and log L/&. Incidentally, the authors have 
312 M. S. Vardya 
cla~med that the present result is the first reliable indication that such an effect actually 
exists in nature. 
The question now is, is there really a discrepancy between results that I obtained and 
those of Nieuwenhuijzen & de Jager (1988) and Howarth & Prlnja (1989) as well as 
theoretical results of Poe & Friend (1986), Friend & Abbott (1986) and others. Perhaps 
not. Apparent d~fferences are due to looking at the problem differently, using different 
kinds of samples, and the problem posed by sin i in the observed data. 
This is dedicated to Professor K. D. Abhyankar on the occasion of his 60th 
birthday. 
References 
Castor, J I , Abbott. D C & Kleln. K K (1975) Ap J 195. 157 
d t  Greve, J P.. de Loore, C & de Jager, C (1972) Ap. Sp. Scr 18, 128 
L)oa7an, V . Franco, M .  L , Stalio, R .  & Thomas, R N.  (1982) I A U  Syrrlp. No. 98. p 318. 
tr~cnd. D B & Abbott. D. C. (1986). Ap J. 311. 701 
tu~enlid. I & Young. A (1980) Ap J ( k t / . )  240, L59 
Galmany. C D .  Olion. G.  L ,  Contr, P S & van Stenberg, M E (1981) Ap J 250, 660 
Cdth~ers, K. .  Lamers, It J Ci L M & Snow, T. P (1981) Ap J. 247, 173 
Howarth, I D & Prlnja, K K (1989) Ap. J. Suppl. 69. 527. 
I:amer5, H .  J G L M .  (1981) Ap J 245. 593 
Marlborough, J .  M M & Zamir, M (1984) A p  J. 276, 706 
Nleuwenhuijzen, H & de Jager, C. (1988) Asrr. Ap. 203, 355. 
Pauldrach, A ,  Puls. J & Kurdr~tzki. R.  P. (1986) Astr Ap. 164. 86. 
Poe. C H & Fr~end, D. B (1986) Ap J. 311, 317. 
Sluttehah, A E & Carpenter, K G (1983) Ap. J. Suppl. 53, 869 
Snow, I'. P (1981, 1982) Ap. J. 251, 139, A p  J. ( L p ~ r . )  253, L39. 
Vardya, M S (1984, 1985) Ap. Sp Scr. 107, 141. Ap. J 299, 255 
Discussion 
Kameswara Rao : Would you comment on the rat~onal for i i ~ ~ l a p i n g  your mass-loss rate 
over the surface or dividing the mass-loss rate divided by surface area'? 
Vardya : The main mechanism of mass loss is radiation pressure, which acts unlformaly 
over the surface. Rotational effects via centrifugal pressure, however, are not uniform 
over the surface. Therefore, averaging over surface works differently in the two cases. 
Hence, it is better to consider mass flux rather than mass loss itself. 
Kameswara Rao : Can you comment on the rational of random distribution of i the 
inclination to the line-of-sight, particularly considering the OB stars which occur in 0 
associations and in clusters which might have a definite orientation for their axis of 
rotation? 
Vardya : Excess of low v stars over random distribution has been attributed to stars in 
associations or clusters having a given inclination. One hopes that if one takes a large 
enough sample from all directions, the sample will be reasonably randomized with 
respect to i. However, to get a complete unbiased fully randomized sample is a difficult 
proposition. 
Rathnasree : Is a similar correlation seen in the rest of the H R  diagram or is it confined 
to OB stars? 
Rotat~on and mass loss m ear1.s type stars 313 
C'ardya : N o  hystcmatic study has been carrled out for stars later than B spectral class As 
mol-e and more ratcs ot mash loss are becolnlng available for stars cooler than B, one can 
look Into thc elfcct  o f  rotation of mahs loss in the other parts of the H-Ii diagram 
Periah : In the CAK theory we encounter negative velocity gradients, and t h ~ s  w~l l  not 
a l l o ~ \ ~  us to procccd any further. Is there any other way out of ~t' '  
Vardya : 1 d o  not k n o w  3 s  I have not used the C A K  theory Ln my work ,  nor have 1 
lookcd Into thc details of ~ t s  computat~onal aspect 
I'eriah : Is i t  not necessary to solve the equations of line transfer, mass Inomentun1 and 
energy c o n s ~ s t c ~ ~ t l y  o obtain mass loss'! 
Vardyn: Ye\. but 11 1s a difficult proposition, bes~dcs, it may not be necessary In all cases, 
cons~dcring thc accuracy of the data 


Rotation and mass loss in early type stars

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Rotation and mass loss in early type stars

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